Math money

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Math Money: A simple introduction to crypto-currencies

Rohas Nagpal
Asian School of Cyber Laws
rn@asianlaws.org

Abstract: Crypto-currencies (bitcoin et al) have caught the attention of
Governments, enforcement agencies, geeks and the general public. This
document provides a simple introduction to crypto-currencies and briefly
introduces terms such as cryptography, hash functions, proof-of-work, digital
signatures, mining, merkle root & tree, crypto-currency addresses and wallets.
This document is intended for the novice reader and may suffer from errors
inherent when a complex topic is (over?) simplified.

Note: Although this document mentions Bitcoin, most of it applies to any system
that “uses public key cryptography, peer-to-peer networking and proof-of-work
to process and verify payments”.

Note: This document is intended for the novice reader and may suffer from
errors inherent when a complex topic is (over?) simplified.

1. Evolution of money: from cowry shells to the blockchain

Our ancestors started off with the barter system - something like "I will give you
2 buffaloes in return for 5 shiny new super-sharp axes". Soon they realised that
the barter system had too many limitations - everyone didn't want buffaloes,
buffaloes were neither divisible (not too many people would want 0.35
buffaloes) nor very portable (imagine having to carry a buffalo on your shoulders
while going shopping).

So they moved on to more acceptable, divisible, homogeneous and portable
forms of money - cowry shells, salt, gold, silver and lots more. The Chinese
invention of paper eventually led to the birth of paper currency, which was
initially backed by gold or other precious metals. Then the world moved on to
fiat money - currency that's declared as legal tender by a government but not
backed by a physical commodity1.

1 Have a look at a 100-rupee note. It caries a promise signed by the Governor of the
Reserve Bank of India (RBI) – “I promise to pay the bearer the sum of one hundred
rupees”. If you were to take this note to the Governor of the RBI, he would give you coins
or one-rupee notes totaling 100 rupees. The RBI gets the power to issue currency notes
by section 31 of the Reserve Bank of India Act, 1934. This section states that “No person
in India other than the Bank or, as expressly authorized by this Act, the Central
Government shall draw, accept, make or issue any bill of exchange, hundi, promissory note
or engagement for the payment of money payable to bearer on demand, or borrow, owe or
take up any sum or sums of money on the bills, hundis or notes payable to bearer on
demand of any such person…”

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This brings us to an essential question – what is money? Money's a matter of
functions four, a Medium, a Measure, a Standard, a Store. So goes the couplet
based on William Stanley Jevons analysis of money in 1875. This meant that for
something to be called as money, it must function as a medium of exchange, a
measure of value, a standard of deferred payment and a store of value.

The birth of computers and the Internet brought in many electronic payment
systems including debit cards, stored value cards, giro transfers, credit cards,
net-banking, electronic bill payments, electronic cheques, mobile wallets, digital
gold currencies, digital wallets, electronic funds transfer at point of sale, mobile
banking, SMS banking, online banking, payment cards, real-time gross settlement
systems, SWIFT, wire transfers and more.

And then came Satoshi Nakamoto’s path breaking whitepaper - Bitcoin: A Peer-
to-Peer Electronic Cash System in October 2008. This brought the world its first
truly peer-to-peer electronic currency2 . Bitcoin earned a lot of notoriety
primarily because of its use by members of the now shut-down Silk Road - an
illegal online marketplace that facilitated the sale of hundreds of millions of
dollars worth of drugs, guns, stolen financial information, counterfeit documents
and more. All Silk Road transactions were conducted exclusively in bitcoin.

A lot of crypto-currencies3 piggybacked on Bitcoin’s underlying innovation – the
blockchain. In fact we now have more than 650 virtual currencies4 being used

2 Virtual currency is a digital representation of value that can be digitally traded and
functions as (1) a medium of exchange; and/or (2) a unit of account; and/or (3) a store
of value, but does not have legal tender status (i.e., when tendered to a creditor, is a
valid and legal offer of payment) in any jurisdiction. It is not issued nor guaranteed by
any jurisdiction, and fulfils the above functions only by agreement within the
community of users of the virtual currency. Virtual currency is distinguished from fiat
currency (a.k.a. “real currency,” “real money,” or “national currency”), which is the coin
and paper money of a country that is designated as its legal tender; circulates; and is
customarily used and accepted as a medium of exchange in the issuing country. It is
distinct from e-money, which is a digital representation of fiat currency used to
electronically transfer value denominated in fiat currency. E-money is a digital transfer
mechanism for fiat currency—i.e., it electronically transfers value that has legal tender
status. [Source: FATF report on Virtual Currencies - Key Definitions and Potential
AML/CFT Risks]

3 Cryptocurrency refers to a math-based, decentralised convertible virtual currency that
is protected by cryptography. - i.e., it incorporates principles of cryptography to
implement a distributed, decentralised, secure information economy. Cryptocurrency
relies on public and private keys to transfer value from one person (individual or entity)
to another, and must be cryptographically signed each time it is transferred. The safety,
integrity and balance of cryptocurrency ledgers is ensured by a network of mutually
distrustful parties (in Bitcoin, referred to as miners) who protect the network in
exchange for the opportunity to obtain a randomly distributed fee (in Bitcoin, a small
number of newly created bitcoins, called the “block reward” and in some cases, also
transaction fees paid by users as a incentive for miners to include their transactions in
the next block). Hundreds of cryptocurrency specifications have been defined, mostly

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around the world. And now we have become a world where bankers wake up
each morning wondering – “has the meaning of money and banking changed while
I slept”.

This rapid change in the global money ecosystem has implications for all of us -
from Governments looking to clamp down on money laundering, tax evasion and
terrorist funding to banks looking to understand the implications of the
blockchain technology. From law enforcement looking to clamp down on the
Mafia using Bitcoin to businesses looking for faster and cheaper ways to receive
and transfer money globally.

2. The mathematics of it all

Sanya’s a naughty young girl who’s been grounded for a week. She wants to
sneak out for desert with her friends but obviously can’t let her dad know about
it. She’s not allowed to use her cellphone, so the only way for her to call her
friends is using the good old landline in her dad’s room.

Since she regularly gets grounded, she and her friends have worked out a simple
system for sharing secrets. When she says, “have you read the book I told you
about” she actually means “let’s sneak out tonight”. When she says something
about “page 10” of the book, she means “pick me up at 10 pm”. Continuing the
logic, page 11 would mean 11 pm and so on.

So on the phone she asks her friend “Have you read the book I told you about?
Page 12 is really funny”, she means, “Let’s sneak out tonight, pick me up at
midnight”.

What we have just seen is cryptography (and a rebellious teenager) in action in
the real world.

The sentence “Let’s sneak out tonight, pick me up at midnight” is plain text –
what Sanya actually wants to convey. The sentence “Have you read the book I
told you about? Page 12 is really funny" is the cipher text – something that an
adversary (her dad in this case) should not be able to understand.

Encryption is the process of converting plain text to cipher text. The reverse
process is decryption.

derived from Bitcoin, which uses a proof- of-work system to validate transactions and
maintain the block chain. While Bitcoin provided the first fully implemented
cryptocurrency protocol, there is growing interest in developing alternative, potentially
more efficient proof methods, such as systems based on proof-of-stake. [Source: FATF
report on Virtual Currencies - Key Definitions and Potential AML/CFT Risks]

4 Source: www.mapofcoins.com, retrieved on 19th March, 2016.

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This science of encrypting and decrypting messages (cryptography) has been
used for thousands of years. It is believed that when Julius Caesar sent messages
to his generals, he replaced every A in his messages with a D, every B with an E,
and so on through the alphabet. Only someone who knew the “shift by 3” rule
could decipher his messages.

For example, if we want to encode the word “SECRET” using Caesar’s key value
of 3, we offset the alphabet so that the 3rd letter down, (D), begins the alphabet.

So starting with ABCDEFGHIJKLMNOPQRSTUVWXYZ
and sliding everything up by 3, you get
DEFGHIJKLMNOPQRSTUVWXYZABC
where D=A, E=B, F=C, and so on.

Using this scheme, the plaintext, “SECRET” encrypts as “VHFUHW”. To allow
someone else to read the cipher text, you tell him or her that the key is 3. This
method is called symmetric cryptography and involves using the same key for
encrypting as well as decrypting a message. This naturally poses a serious
problem – what if an adversary gets hold of this key? At some point of time the
sender and receiver need to exchange the key. That’s when an adversary could
get hold of the key. In modern cryptography, keys are really really large
numbers.

The secure-key-exchange problem was solved with the birth of asymmetric key
cryptography – in which two different but related keys are used - the public key
to encrypt data and the corresponding private key to decrypt the data. If Sanya
were to send an encrypted message to Karan, she would encrypt the message
using his public key (which is available to the world). Once encrypted, the
message can only be decrypted using Karan’s private key (which would only be
available to Karan).

To understand how this works, lets look at the RSA algorithm (named after its
inventors Ron Rivest, Adi Shamir, and Leonard Adleman).

The RSA public-key encryption algorithm works in the following manner:
1. Generation of a public-private key pair.
2. Encryption of a message (plain text) with the public key generated in step
(1) to get the cipher-text.
3. Decryption of the cipher-text by using the corresponding private key
generated in step (1).

Step 1: Generation of a key pair
1. Select two large integer primes p and q.
2. Multiply p and q to get a number n, that means, pq = n.
3. Obtain φ which is the product of (p-1) and (q-1), that means
φ = (p-1)(q-1).

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4. Select e such that 1<e<φ and the greatest common divisor of e and φ is 1.
That means e and φ are coprime.
5. Compute d such that 1<d<φ and ed ≡ 1 mod φ. This means that the value
of d must be such that ed-1 should be completely divisible by φ or
(ed-1) / φ should be an integer.
6. The public-key is (e, n) and the corresponding private key is (d, n).

Step 2: Encryption process
Suppose the message to be encrypted is m. The cipher-text c is obtained by
raising the message to the value of e and finding out its modulo n.

That means
c = me mod n.

Step 3: Decryption process
Decryption is achieved by raising the cipher-text c obtained in step 2 to the value
of d and finding out its modulo n.

That means m=cd mod n.

Let’s try the algorithm with really small prime numbers5: 3 and 11. (In reality
the primes chosen would be really really large).

1. Choose p = 3 and q = 11

2. Compute n = p * q = 3 * 11 = 33

3. Compute φ = (p - 1) * (q - 1) = 2 * 10 = 20

4. Choose e such that 1 < e < φ and e and φ are coprime. Let e = 7

5. Compute a value for d such that 1<d<φ and ed ≡ 1 mod φ.
One solution is d = 3.

6. Public key is (e, n) => (7, 33)
Private key is (d, n) => (3, 33)

7. Suppose the plain text is 2.
The cipher text will be c = me mod n.
That’s 27 mod 33 = 128 mod 33 = 29

8. The decryption will be
cd mod n

5 Source: https://www.cs.utexas.edu/~mitra/honors/soln.html

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= 293 mod 33
= 24389 mod 33
= 2

The security of the RSA cryptosystem is based on the integer factorization
problem. Any adversary who wishes to decipher the cipher-text c must do so by
using the publicly available information (n, e). One possible method is to first
factor n, and then compute φ and d just as was done in the above mentioned
steps. The factoring of n is currently computationally infeasible (provided
sufficiently large prime numbers are chosen as p and q) and therein lays the
strength of the RSA cryptosystem.

Before we get into the nuts and bolts of how crypto-currencies work, we need to
understand some more concepts including hash functions. A one-way hash
function takes an input (e.g. a PDF file, a video, an email, a string etc.) and
produces a fixed-length output e.g. 160-bits.

The hash function ensures that if the information is changed in any way – even
by just one bit – an entirely different output value is produced. The table below
shows some sample output values using the sha1 (40) hash function6.

Input Hash
sanya c75491c89395de9fa4ed29affda0e4d29cbad290
SANYA 33fef490220a0e6dee2f16c5a8f78ce491741adc
Sanya 4c391643f247937bee14c0bcca9ffb985fc0d0ba

It can be seen from the table above that by changing the input from sanya to
SANYA, an entirely different hash value is generated. What must be kept in mind
is that irrespective of the size of the input, the hash output will always be of the
same size.

Two things must be borne in mind with regard to one-way hash functions:
1. It is computationally infeasible to find two different input messages that
will yield the same hash output.
2. It is computationally infeasible to reconstruct the original message from
its hash output.

Having understood hash functions, let’s have a look at another interesting
concept called proof-of-work. This is a way to reduce spam and denial of service
attacks by requiring a computer to spend some time and processing power to
solve something.

One such proof-of-work system that is used in crypto-currencies is hashcash. The
basic premise of hashcash is that if the sender of an email can prove that she has

6 Computing hash of an electronic record is a very simple process. E.g. in php it can be
done with: hash_file('sha256', $filename).

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spent reasonable time and computational power to solve some puzzle, it can be
believed that the sender is not a spammer. The logic is that spamming would be
economically infeasible if a spammer had to spend non-trivial time and
computational power for every single email being sent.

Let’s develop an elementary proof-of-work system, based on hashcash, which
can be used to control spam. Let’s presume that rn@asianlaws.org is sending an
email to info@lexcode.com. The sender must include something similar to the
following in the header of the email:

rn@asianlaws.org: info@lexcode.com:18032016:xxxx

That’s 4 pieces of information separated by colons. The first piece is the sender’s
email address, the second is the receiver’s email address and the third is the
current date in DDMMYYYY format. The fourth piece is something that needs to
be calculated by the sender’s computer. Let’s call it a nonce.

The objective is to find an input that would result in a sha256 hash which begins
with 5 zeros.

So we start the nonce at a value of 0 and then keep incrementing it (1, 2, 3 … )
and calculating the hash. Something like this:

Input rn@asianlaws.org:info@lexcode.com:18032016:1
sha256 hash 288721860bec3a490811981c831702d4f41e54c3f8c183c5650ac73ff231659c

sha256 hash 11caf434535c35cdc843e801382f0a8643a03500649a9bfa41c8e6a4be65a413

sha256 hash aad80b9c58e977a5da90f81b2667af443b50425876920528f237df0a6ffe1aa4

And so on till .. 1580661

sha256 hash 0000080602f705257e74a4e847e9ed23ab61be5b2ba4263fbacc90bd7c7c7ab4

Calculating this may not take a genuine sender a lot of time and computational
power but if a spammer were to make these calculations for millions of emails, it
will take a non-trivial amount of time and computational power.

At the receiver’s end, the computer will simply take the following line from the
header of the email and calculate the hash.

rn@asianlaws.org:info@lexcode.com:18032016:1580661

If the hash begins with a pre-defined number of zeros (5 in this example), the
email would not be considered spam. This will take the receiver a trivial amount
of time and computational power since it just has to calculate the hash of one

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input. The date can be used as an additional validation parameter – e.g. if the
date is within 24 hours of the time of receipt, the email will be approved for
download.

A very important application of public key cryptography is a digital signature.
In this, the signer first calculates the hash of the message she wants to digitally
sign. Then using her private key and the hash, she creates a digital signature,
using the relevant algorithm.

This digital signature is unique to the message.

The signer then sends the message and the digital signature to the receiver. The
receiver re-computes the hash from the message. The receiver also computes
another string using the digital signature and the signer’s public key (using the
relevant algorithm). If this string and the hash match, the digital signature is
verified.

Blind digital signatures were subsequently developed for use in digital cash
and cryptographic voting systems. In this system, the content of the message is
disguised before it is signed. The resulting blind signature can be verified against
the original, un-blinded message in the manner of a regular digital signature7.

However, blind digital signatures do not solve the double-spending problem. In
case of physical currency notes, you cannot double-spend a note because once
you hand the note over to someone, you don’t have the note anymore to spend
again. Since electronic records are easily duplicated, a “digital coin” can be spent
multiple times.

Bitcoin solves the double-spending problem through the blockchain - a public
ledger containing an ordered and time-stamped record of transactions. In
addition to preventing double-spending, the blockchain prevents the
modification of previous transaction records.

A block of one or more new transactions is collected into the transaction data
part of a block. Copies of each transaction are hashed, and the hashes are then

7 Source: https://en.wikipedia.org/wiki/Blind_signature. An often-used analogy to the
cryptographic blind signature is the physical act of a voter enclosing a completed
anonymous ballot in a special carbon paper lined envelope that has the voter's
credentials pre-printed on the outside. The ballot can be marked through the envelope
by the carbon paper. The voter hands the sealed envelope to an official who verifies the
credentials and signs it. Once signed, the package is given back to the voter, who
transfers the now signed ballot to a new unmarked normal envelope. Thus, the signer
does not view the message content, but a third party can later verify the signature and
know that the signature is valid within the limitations of the underlying signature
scheme.

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paired, hashed, paired again, and hashed again until a single hash remains, the
merkle root of a merkle tree8.

Lets consider a simple illustration of how the blockchain works. Consider a block
that has 6 transactions a, b, c, d, e and f.

The merkle tree is:
d1 = double-hash (a)
d2 = double-hash (b)
d3 = double-hash (c)
d4 = double-hash (d)
d5 = double-hash (e)
d6 = double-hash (f)

d7 = double-hash (d1 concatenated with d2)

Since there are an odd number of hashes, we take d9 twice


d12 is the merkle root of the 6 transactions in this block. This is stored in the
block header. Additionally, each block also stores the hash of the header of the
previous block. This chains the blocks together and ensures that a transaction
cannot be modified without modifying the block that records it and all following
blocks. Transactions are also chained together.

Bitcoin uses a proof-of-work technique similar (but more complex) than the one
discussed earlier in this document. Since good cryptographic hash algorithms
convert arbitrary inputs into “seemingly-random” hashes, it is not feasible to
modify the input to make the hash predictable. To prove that she did some extra
work to create a block, a miner must create a hash of the block header, which
does not exceed a certain value.

The term miner must not be compared with a gold or coal miner in the real
world. While a gold miner digs into the earth to discover gold, a bitcoin miner
uses computational power to calculate hashes. To add an entire block to the

8 Source: https://bitcoin.org/en/developer-guide#block-chain-overview

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block chain, a Bitcoin miner must successfully hash a block header to a value
below the target threshold. Bitcoin miners spend a lot of money (for
computational power and electricity) and are compensated by way of a reward
for each block they mine – this was initially 50 bitcoins per block and is halving
every 210,000 blocks. Miners also earn transaction fees. Miners usually operate
as part of a large pool instead of as individuals.

Interestingly, Bitcoins can be also be mined with a pencil and paper9.

The first-ever Bitcoin block is known as the genesis block. Each subsequent
block is addressed by its block height, which represents the number of blocks
between it and the genesis block.

New blocks are added to the block chain if their hash is at least as challenging as
a difficulty value expected by the Bitcoin consensus protocol. According to the
bitcoin protocol, it should take 2 weeks for 2016 blocks to be generated. If the
time taken is more or less than 2 weeks then the difficulty value is relatively
decreased or increased.

The overview of the Bitcoin process is explained very well on Ken Shirriff's
blog10.

9 See: http://www.righto.com/2014/09/mining-bitcoin-with-pencil-and-paper.html
10 To simplify slightly, bitcoins consist of entries in a distributed database that keeps
track of the ownership of bitcoins. Unlike a bank, bitcoins are not tied to users or
accounts. Instead bitcoins are owned by a Bitcoin address, for example
1KKKK6N21XKo48zWKuQKXdvSsCf95ibHFa. A transaction is the mechanism for
spending bitcoins. In a transaction, the owner of some bitcoins transfers ownership to a
new address. A key innovation of Bitcoin is how transactions are recorded in the
distributed database through mining. Transactions are grouped into blocks and about
every 10 minutes a new block of transactions is sent out, becoming part of the
transaction log known as the blockchain, which indicates the transaction has been made
(more-or-less) official. Bitcoin mining is the process that puts transactions into a block,
to make sure everyone has a consistent view of the transaction log. To mine a block,
miners must find an extremely rare solution to an (otherwise-pointless) cryptographic
problem. Finding this solution generates a mined block, which becomes part of the
official block chain.
Mining is also the mechanism for new bitcoins to enter the system. When a block
is successfully mined, new bitcoins are generated in the block and paid to the miner.
This mining bounty is large - currently 25 bitcoins per block. In addition, the miner gets
any fees associated with the transactions in the block. Because of this, mining is very
competitive with many people attempting to mine blocks. The difficulty and
competitiveness of mining is a key part of Bitcoin security, since it ensures that nobody
can flood the system with bad blocks.
There is no centralized Bitcoin server. Instead, Bitcoin runs on a peer-to-peer
network. If you run a Bitcoin client, you become part of that network. The nodes on the
network exchange transactions, blocks, and addresses of other peers with each other.
When you first connect to the network, your client downloads the blockchain from some
random node or nodes. In turn, your client may provide data to other nodes. When you
create a Bitcoin transaction, you send it to some peer, who sends it to other peers, and
so on, until it reaches the entire network. Miners pick up your transaction, generate a

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A Bitcoin address is an identifier of 26 to 35 alphanumeric characters, beginning
with the number 1 or 3, which represents a possible destination for a bitcoin
payment. Addresses can be generated at no cost by any user of Bitcoin11.

mined block containing your transaction, and send this mined block to peers. Eventually
your client will receive the block and your client shows that the transaction was
processed.
Bitcoin uses digital signatures to ensure that only the owner of bitcoins can
spend them. The owner of a Bitcoin address has the private key associated with the
address. To spend bitcoins, they sign the transaction with this private key, which proves
they are the owner. (It's somewhat like signing a physical check to make it valid.) A
public key is associated with each Bitcoin address, and anyone can use it to verify the
digital signature.
Blocks and transactions are identified by a 256-bit cryptographic hash of their
contents. This hash value is used in multiple places in the Bitcoin protocol. In addition,
finding a special hash is the difficult task in mining a block.

11 Source: https://en.bitcoin.it/wiki/Address. The technical process to create Bitcoin
addresses is explained as under:
• Step 0: Having a private Elliptic Curve Digital Signature Algorithm key:
18E14A7B6A307F426A94F8114701E7C8E774E7F9A47E2C2035DB29A206321725
• Step 1: Take the corresponding public key generated with it:
0450863AD64A87AE8A2FE83C1AF1A8403CB53F53E486D8511DAD8A04887E5B2352
2CD470243453A299FA9E77237716103ABC11A1DF38855ED6F2EE187E9C582BA6
• Step 2: Perform SHA-256 hashing on the public key
600FFE422B4E00731A59557A5CCA46CC183944191006324A447BDB2D98D4B408
• Step 3: Perform RIPEMD-160 hashing on the result of SHA-256
010966776006953D5567439E5E39F86A0D273BEE
• Step 4: Add version byte in front of RIPEMD-160 hash (0x00 for Main Network)
00010966776006953D5567439E5E39F86A0D273BEE
(note that below steps are the Base58Check encoding11)
• Step 5: Perform SHA-256 hash on the extended RIPEMD-160 result
445C7A8007A93D8733188288BB320A8FE2DEBD2AE1B47F0F50BC10BAE845C094
• Step 6: Perform SHA-256 hash on the result of the previous SHA-256 hash
D61967F63C7DD183914A4AE452C9F6AD5D462CE3D277798075B107615C1A8A30
• Step 7: Take the first 4 bytes of the second SHA-256 hash. This is the address
checksum
D61967F6
• Step 8: Add the 4 checksum bytes from the previous stage at the end of extended
RIPEMD-160 hash from stage 4. This is the 25-byte binary Bitcoin Address.
00010966776006953D5567439E5E39F86A0D273BEED61967F6
• Step 9: Convert the result from a byte string into a base58 string using
Base58Check encoding. This is the most commonly used Bitcoin Address format
16UwLL9Risc3QfPqBUvKofHmBQ7wMtjvM

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There are currently two address formats in common use:

Common P2PKH which begin with the
number 1
Example:
1BvBMSEYstWetqTFn5Au4m4GFg7xJaNVN2

Newer P2SH type starting with the
number 3
Example:
3J98t1WpEZ73CNmQviecrnyiWrnqRhWNLy

Bitcoin wallets at their core are a collection of private keys.

These collections are stored digitally in a file, or can even be physically stored on
pieces of paper.

The simplest Bitcoin wallet is a program, which performs these functions12:
• it generates private keys,
• derives the corresponding public keys,
• helps distribute those public keys as necessary,
• monitors for outputs spent to those public keys,
• creates and signs transactions spending those outputs, and
• broadcasts the signed transactions.

Although it’s called a wallet, a Bitcoin wallet does not store bitcoins. The wallet is
a collection of public-private key-pairs.

As discussed, the blockchain is a database of transaction information. It is
constantly growing and is sent out to all nodes in the Bitcoin network. Every
transaction is distributed to the network and all valid transactions are included
in the next block, which is mined.

Imagine a real-world transaction where your salary is transferred to your bank
account through an online transfer made by your employer. You then use your
debit card to pay for dinner. This transfers some of the money to the restaurant’s
account. In these 2 transactions, did you see a single currency note? No. So we
can say that in today’s world most money exists as a history of transactions and
balances.

Bitcoin, or for that matter most virtual currencies work the same way. They don’t
actually “exist” in the true sense of the word. They just are there!

A bitcoin can be divided down to 8 decimal places - 0.00000001 is the smallest
amount, also referred to as a satoshi. The last block that will generate bitcoins
will be block 6,929,999. This is expected to be generated around the year 2140.
After that, the total number of bitcoins will remain static at just below 21 million.

12 https://bitcoin.org/en/developer-guide#wallet-programs

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3. Additional reading and other resources

1. Blind Digital Signatures
http://www.hit.bme.hu/~buttyan/courses/BMEVIHIM219/2009/Chaum.BlindSigForPa
yment.1982.PDF

2. Bitcoin: A Peer-to-Peer Electronic Cash System
https://bitcoin.org/bitcoin.pdf

3. Bitcoin wiki
https://en.bitcoin.it/wiki/

4. FATF Report on Virtual Currencies Key Definitions and Potential AML/CFT Risks
http://www.fatf-gafi.org/media/fatf/documents/reports/Virtual-currency-key-
definitions-and-potential-aml-cft-risks.pdf

4. License

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International
License. You are free to:
Share — copy and redistribute the material in any medium or format
Adapt — remix, transform, and build upon the material for any purpose, even commercially.

The licensor cannot revoke these freedoms as long as you follow the license terms.

Under the following terms:
Attribution — You must give appropriate credit, provide a link to the license, and indicate if
changes were made. You may do so in any reasonable manner, but not in any way that suggests
the licensor endorses you or your use.
ShareAlike — If you remix, transform, or build upon the material, you must distribute your
contributions under the same license as the original.
No additional restrictions — You may not apply legal terms or technological measures that
legally restrict others from doing anything the license permits.

Notices:
You do not have to comply with the license for elements of the material in the public domain or
where your use is permitted by an applicable exception or limitation.
No warranties are given. The license may not give you all of the permissions necessary for your
intended use. For example, other rights such as publicity, privacy, or moral rights may limit how
you use the material.

Document version 1.0 dated 20th March, 2016.

Math money

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